Theorem madebdayim 27260

Description: If a surreal is a member of a made set, its birthday is less than or equal to the level. (Contributed by Scott Fenton, 10-Aug-2024.)

Assertion

Ref	Expression
madebdayim	⊢ (𝑋 ∈ ( M ‘𝐴) → ( bday ‘𝑋) ⊆ 𝐴)

Proof of Theorem madebdayim

Dummy variables 𝑎 𝑏 𝑥 𝑦 𝑧 𝑙 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvdm 6884	. . 3 ⊢ (𝑋 ∈ ( M ‘𝐴) → 𝐴 ∈ dom M )
2		madef 27229	. . . 4 ⊢ M :On⟶𝒫 No
3	2	fdmi 6685	. . 3 ⊢ dom M = On
4	1, 3	eleqtrdi 2842	. 2 ⊢ (𝑋 ∈ ( M ‘𝐴) → 𝐴 ∈ On)
5		fveq2 6847	. . . . . 6 ⊢ (𝑎 = 𝑏 → ( M ‘𝑎) = ( M ‘𝑏))
6		sseq2 3973	. . . . . 6 ⊢ (𝑎 = 𝑏 → (( bday ‘𝑥) ⊆ 𝑎 ↔ ( bday ‘𝑥) ⊆ 𝑏))
7	5, 6	raleqbidv 3317	. . . . 5 ⊢ (𝑎 = 𝑏 → (∀𝑥 ∈ ( M ‘𝑎)( bday ‘𝑥) ⊆ 𝑎 ↔ ∀𝑥 ∈ ( M ‘𝑏)( bday ‘𝑥) ⊆ 𝑏))
8		fveq2 6847	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ( bday ‘𝑥) = ( bday ‘𝑦))
9	8	sseq1d 3978	. . . . . 6 ⊢ (𝑥 = 𝑦 → (( bday ‘𝑥) ⊆ 𝑏 ↔ ( bday ‘𝑦) ⊆ 𝑏))
10	9	cbvralvw 3223	. . . . 5 ⊢ (∀𝑥 ∈ ( M ‘𝑏)( bday ‘𝑥) ⊆ 𝑏 ↔ ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏)
11	7, 10	bitrdi 286	. . . 4 ⊢ (𝑎 = 𝑏 → (∀𝑥 ∈ ( M ‘𝑎)( bday ‘𝑥) ⊆ 𝑎 ↔ ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏))
12		fveq2 6847	. . . . 5 ⊢ (𝑎 = 𝐴 → ( M ‘𝑎) = ( M ‘𝐴))
13		sseq2 3973	. . . . 5 ⊢ (𝑎 = 𝐴 → (( bday ‘𝑥) ⊆ 𝑎 ↔ ( bday ‘𝑥) ⊆ 𝐴))
14	12, 13	raleqbidv 3317	. . . 4 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ ( M ‘𝑎)( bday ‘𝑥) ⊆ 𝑎 ↔ ∀𝑥 ∈ ( M ‘𝐴)( bday ‘𝑥) ⊆ 𝐴))
15		elmade2 27241	. . . . . . . 8 ⊢ (𝑎 ∈ On → (𝑥 ∈ ( M ‘𝑎) ↔ ∃𝑙 ∈ 𝒫 ( O ‘𝑎)∃𝑟 ∈ 𝒫 ( O ‘𝑎)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)))
16	15	adantr 481	. . . . . . 7 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → (𝑥 ∈ ( M ‘𝑎) ↔ ∃𝑙 ∈ 𝒫 ( O ‘𝑎)∃𝑟 ∈ 𝒫 ( O ‘𝑎)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥)))
17		elpwi 4572	. . . . . . . . . . 11 ⊢ (𝑙 ∈ 𝒫 ( O ‘𝑎) → 𝑙 ⊆ ( O ‘𝑎))
18		elpwi 4572	. . . . . . . . . . 11 ⊢ (𝑟 ∈ 𝒫 ( O ‘𝑎) → 𝑟 ⊆ ( O ‘𝑎))
19	17, 18	anim12i 613	. . . . . . . . . 10 ⊢ ((𝑙 ∈ 𝒫 ( O ‘𝑎) ∧ 𝑟 ∈ 𝒫 ( O ‘𝑎)) → (𝑙 ⊆ ( O ‘𝑎) ∧ 𝑟 ⊆ ( O ‘𝑎)))
20		unss 4149	. . . . . . . . . 10 ⊢ ((𝑙 ⊆ ( O ‘𝑎) ∧ 𝑟 ⊆ ( O ‘𝑎)) ↔ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎))
21	19, 20	sylib 217	. . . . . . . . 9 ⊢ ((𝑙 ∈ 𝒫 ( O ‘𝑎) ∧ 𝑟 ∈ 𝒫 ( O ‘𝑎)) → (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎))
22		simpr 485	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → 𝑙 <<s 𝑟)
23		simplll 773	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → 𝑎 ∈ On)
24		dfss3 3935	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎) ↔ ∀𝑧 ∈ (𝑙 ∪ 𝑟)𝑧 ∈ ( O ‘𝑎))
25		fveq2 6847	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑧 → ( bday ‘𝑦) = ( bday ‘𝑧))
26	25	sseq1d 3978	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = 𝑧 → (( bday ‘𝑦) ⊆ 𝑏 ↔ ( bday ‘𝑧) ⊆ 𝑏))
27	26	rspccv 3579	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏 → (𝑧 ∈ ( M ‘𝑏) → ( bday ‘𝑧) ⊆ 𝑏))
28	27	ralimi 3082	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏 → ∀𝑏 ∈ 𝑎 (𝑧 ∈ ( M ‘𝑏) → ( bday ‘𝑧) ⊆ 𝑏))
29		rexim 3086	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑏 ∈ 𝑎 (𝑧 ∈ ( M ‘𝑏) → ( bday ‘𝑧) ⊆ 𝑏) → (∃𝑏 ∈ 𝑎 𝑧 ∈ ( M ‘𝑏) → ∃𝑏 ∈ 𝑎 ( bday ‘𝑧) ⊆ 𝑏))
30	28, 29	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏 → (∃𝑏 ∈ 𝑎 𝑧 ∈ ( M ‘𝑏) → ∃𝑏 ∈ 𝑎 ( bday ‘𝑧) ⊆ 𝑏))
31	30	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → (∃𝑏 ∈ 𝑎 𝑧 ∈ ( M ‘𝑏) → ∃𝑏 ∈ 𝑎 ( bday ‘𝑧) ⊆ 𝑏))
32		elold 27242	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ On → (𝑧 ∈ ( O ‘𝑎) ↔ ∃𝑏 ∈ 𝑎 𝑧 ∈ ( M ‘𝑏)))
33	32	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → (𝑧 ∈ ( O ‘𝑎) ↔ ∃𝑏 ∈ 𝑎 𝑧 ∈ ( M ‘𝑏)))
34		bdayelon 27159	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ( bday ‘𝑧) ∈ On
35		onelssex 6370	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((( bday ‘𝑧) ∈ On ∧ 𝑎 ∈ On) → (( bday ‘𝑧) ∈ 𝑎 ↔ ∃𝑏 ∈ 𝑎 ( bday ‘𝑧) ⊆ 𝑏))
36	34, 35	mpan 688	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 ∈ On → (( bday ‘𝑧) ∈ 𝑎 ↔ ∃𝑏 ∈ 𝑎 ( bday ‘𝑧) ⊆ 𝑏))
37	36	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → (( bday ‘𝑧) ∈ 𝑎 ↔ ∃𝑏 ∈ 𝑎 ( bday ‘𝑧) ⊆ 𝑏))
38	31, 33, 37	3imtr4d 293	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → (𝑧 ∈ ( O ‘𝑎) → ( bday ‘𝑧) ∈ 𝑎))
39	38	ralimdv 3162	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → (∀𝑧 ∈ (𝑙 ∪ 𝑟)𝑧 ∈ ( O ‘𝑎) → ∀𝑧 ∈ (𝑙 ∪ 𝑟)( bday ‘𝑧) ∈ 𝑎))
40	24, 39	biimtrid 241	. . . . . . . . . . . . . . . 16 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → ((𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎) → ∀𝑧 ∈ (𝑙 ∪ 𝑟)( bday ‘𝑧) ∈ 𝑎))
41	40	imp 407	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) → ∀𝑧 ∈ (𝑙 ∪ 𝑟)( bday ‘𝑧) ∈ 𝑎)
42	41	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → ∀𝑧 ∈ (𝑙 ∪ 𝑟)( bday ‘𝑧) ∈ 𝑎)
43		bdayfun 27155	. . . . . . . . . . . . . . . . 17 ⊢ Fun bday
44		oldssno 27234	. . . . . . . . . . . . . . . . . . 19 ⊢ ( O ‘𝑎) ⊆ No
45		sstr 3955	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎) ∧ ( O ‘𝑎) ⊆ No ) → (𝑙 ∪ 𝑟) ⊆ No )
46	44, 45	mpan2 689	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎) → (𝑙 ∪ 𝑟) ⊆ No )
47		bdaydm 27157	. . . . . . . . . . . . . . . . . 18 ⊢ dom bday = No
48	46, 47	sseqtrrdi 3998	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎) → (𝑙 ∪ 𝑟) ⊆ dom bday )
49		funimass4 6912	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun bday ∧ (𝑙 ∪ 𝑟) ⊆ dom bday ) → (( bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎 ↔ ∀𝑧 ∈ (𝑙 ∪ 𝑟)( bday ‘𝑧) ∈ 𝑎))
50	43, 48, 49	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ ((𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎) → (( bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎 ↔ ∀𝑧 ∈ (𝑙 ∪ 𝑟)( bday ‘𝑧) ∈ 𝑎))
51	50	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) → (( bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎 ↔ ∀𝑧 ∈ (𝑙 ∪ 𝑟)( bday ‘𝑧) ∈ 𝑎))
52	51	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → (( bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎 ↔ ∀𝑧 ∈ (𝑙 ∪ 𝑟)( bday ‘𝑧) ∈ 𝑎))
53	42, 52	mpbird 256	. . . . . . . . . . . . 13 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → ( bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎)
54		scutbdaybnd 27197	. . . . . . . . . . . . 13 ⊢ ((𝑙 <<s 𝑟 ∧ 𝑎 ∈ On ∧ ( bday “ (𝑙 ∪ 𝑟)) ⊆ 𝑎) → ( bday ‘(𝑙 |s 𝑟)) ⊆ 𝑎)
55	22, 23, 53, 54	syl3anc 1371	. . . . . . . . . . . 12 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → ( bday ‘(𝑙 |s 𝑟)) ⊆ 𝑎)
56		fveq2 6847	. . . . . . . . . . . . 13 ⊢ ((𝑙 |s 𝑟) = 𝑥 → ( bday ‘(𝑙 |s 𝑟)) = ( bday ‘𝑥))
57	56	sseq1d 3978	. . . . . . . . . . . 12 ⊢ ((𝑙 |s 𝑟) = 𝑥 → (( bday ‘(𝑙 |s 𝑟)) ⊆ 𝑎 ↔ ( bday ‘𝑥) ⊆ 𝑎))
58	55, 57	syl5ibcom 244	. . . . . . . . . . 11 ⊢ ((((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) ∧ 𝑙 <<s 𝑟) → ((𝑙 |s 𝑟) = 𝑥 → ( bday ‘𝑥) ⊆ 𝑎))
59	58	expimpd 454	. . . . . . . . . 10 ⊢ (((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) ∧ (𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎)) → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → ( bday ‘𝑥) ⊆ 𝑎))
60	59	ex 413	. . . . . . . . 9 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → ((𝑙 ∪ 𝑟) ⊆ ( O ‘𝑎) → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → ( bday ‘𝑥) ⊆ 𝑎)))
61	21, 60	syl5 34	. . . . . . . 8 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → ((𝑙 ∈ 𝒫 ( O ‘𝑎) ∧ 𝑟 ∈ 𝒫 ( O ‘𝑎)) → ((𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → ( bday ‘𝑥) ⊆ 𝑎)))
62	61	rexlimdvv 3200	. . . . . . 7 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → (∃𝑙 ∈ 𝒫 ( O ‘𝑎)∃𝑟 ∈ 𝒫 ( O ‘𝑎)(𝑙 <<s 𝑟 ∧ (𝑙 |s 𝑟) = 𝑥) → ( bday ‘𝑥) ⊆ 𝑎))
63	16, 62	sylbid 239	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → (𝑥 ∈ ( M ‘𝑎) → ( bday ‘𝑥) ⊆ 𝑎))
64	63	ralrimiv 3138	. . . . 5 ⊢ ((𝑎 ∈ On ∧ ∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏) → ∀𝑥 ∈ ( M ‘𝑎)( bday ‘𝑥) ⊆ 𝑎)
65	64	ex 413	. . . 4 ⊢ (𝑎 ∈ On → (∀𝑏 ∈ 𝑎 ∀𝑦 ∈ ( M ‘𝑏)( bday ‘𝑦) ⊆ 𝑏 → ∀𝑥 ∈ ( M ‘𝑎)( bday ‘𝑥) ⊆ 𝑎))
66	11, 14, 65	tfis3 7799	. . 3 ⊢ (𝐴 ∈ On → ∀𝑥 ∈ ( M ‘𝐴)( bday ‘𝑥) ⊆ 𝐴)
67		fveq2 6847	. . . . 5 ⊢ (𝑥 = 𝑋 → ( bday ‘𝑥) = ( bday ‘𝑋))
68	67	sseq1d 3978	. . . 4 ⊢ (𝑥 = 𝑋 → (( bday ‘𝑥) ⊆ 𝐴 ↔ ( bday ‘𝑋) ⊆ 𝐴))
69	68	rspccv 3579	. . 3 ⊢ (∀𝑥 ∈ ( M ‘𝐴)( bday ‘𝑥) ⊆ 𝐴 → (𝑋 ∈ ( M ‘𝐴) → ( bday ‘𝑋) ⊆ 𝐴))
70	66, 69	syl 17	. 2 ⊢ (𝐴 ∈ On → (𝑋 ∈ ( M ‘𝐴) → ( bday ‘𝑋) ⊆ 𝐴))
71	4, 70	mpcom 38	1 ⊢ (𝑋 ∈ ( M ‘𝐴) → ( bday ‘𝑋) ⊆ 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3060  ∃wrex 3069   ∪ cun 3911   ⊆ wss 3913  𝒫 cpw 4565   class class class wbr 5110  dom cdm 5638   “ cima 5641  Oncon0 6322  Fun wfun 6495  ‘cfv 6501  (class class class)co 7362   No csur 27025   bday cbday 27027   <<s csslt 27163   |s cscut 27165   M cmade 27215   O cold 27216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-1o 8417  df-2o 8418  df-no 27028  df-slt 27029  df-bday 27030  df-sslt 27164  df-scut 27166  df-made 27220  df-old 27221

This theorem is referenced by:  oldbdayim  27261  madebday  27272